Take it to your nearest tailor and they will give you a quote. I have only ever had zips replaced (about 15 each!) if the outside is still nice it could be your chance to choose some funky lining fabric and update it.
Well we have a couple of tailors in town...they are sometimes a part of a shoe menders, or near a station for some strange reason. But I would look under clothes alterations. bear in mind even if it costs you £40 you will effectively be getting a 'new' coat. You couldn't replace it for that much.
Depends how much you want to save the coat. I can never find just the right long black winter coat so had my faithful Karen Millen one re-lined for £120....I think you could get it done cheaper though. That was a combination of choosing gorgeous teal satin and using a tailor in the middle of the City of London!